Typically, when there's news about the rivalry to find the Higgs, we often see this rivalry between the LHC at CERN and the Tevatron at Fermilab. But in this case, it is the "internal rivalry" between two giant detectors at the LHC - ATLAS and CMS. The article describes the friendly rivalry between these two detector groups, and it also has a short video of it.
Be it a rivalry or not, both detectors need each other (and other detectors situated at the LHC) to verify what the other discovers. This is no different than the situation at the Tevatron with CDF and DZero. They may have a friendly rivalry, but in the end, one needs the other to validate discoveries and measurements. The rivalry serves to keep each other honest and on their toes, which is good for science.